📘 Session 3.6 - Geometric & Negative Binomial Distributions

🎯 Geometric Distribution

Definition: Number of trials until the first success in independent, identical trials with constant success probability p.

PMF: P(X = k) = (1 - p)^(k - 1) · p, for k = 1,2,...
CDF: P(X ≤ k) = 1 - (1 - p)^k
Mean: μ = 1/p
Variance: σ² = (1 - p)/p²

Why the PMF? To get the first success on trial k: fail the first k−1 trials, then succeed → (1-p)^(k-1) · p.

Examples: Electronics QA (first part that meets spec), Mechanical testing (first bearing passing stress), Student life (first successful Wi-Fi login).

🧮 Interactive Calculator — Geometric

p (probability of success) Check probability at k (marks k on the charts)

🔎 Standard geometric (constant p): The graph depends only on p, not on k. The k box is for computing P(X=k) and P(X≤k) and for marking k on the charts.

Optional: Learning mode — let success probability rise with experience (not a geometric model)

Enable learning mode Model: p_k = min(1, p0 + g·(k−1))

When learning mode is on, we compute P(X=k)=product(1-p_i) · p_k and approximate mean/variance from the plotted distribution.

🎯 Negative Binomial Distribution

Definition: Number of trials until the r-th success in independent, identical trials with constant success probability p. (Geometric is the special case r = 1.)

PMF: P(X = k) = C(k-1, r-1) · (1 - p)^(k - r) · p^r, for k = r, r+1,...
CDF: sum of PMFs up to k
Mean: μ = r / p
Variance: σ² = r * (1 - p) / p^2

Interpretation tip: First decide what counts as a success — a pass, a good unit, or a defect (if you are counting defects). X is the number of trials; it tells you how many trials it takes to reach r successes.

Example sequence (r = 3): F, S, F, S, S → the 3rd success is on trial 5, so X = 5. Here k is the trial where the r-th success occurs (so k ≥ r).

🔌 Electrical / Computer Engineering examples

Packet transmissions: Success = “correct decode.” X = packets sent until the r-th correct decode.
Bit-error testing: Success = “bit error occurs” (independent errors). X = bit index of the r-th error.
Sensor reads: Success = “valid reading.” X = readings until r valid ones.
Component QA: Success = “board passes test.” X = units until r passes.

⚙️ Mechanical / Manufacturing examples

Tolerance checks: Success = “within tolerance.” X = parts until the r-th in-tolerance part.
Torque testing: Success = “meets spec.” X = trials until the r-th pass.
Defect sampling: Success = “defective.” X = items checked until the r-th defect.
Assembly QC: Success = “final inspection pass.” X = units until the r-th pass.

🧮 Interactive Calculator — Negative Binomial

r (number of successes) p (success probability) Check probability at k (marks k on the charts)

NB shape depends on r and p. The k box computes P(X=k), P(X≤k), and marks k.

📎 Excel Cheat Sheet

Distribution	PMF	CDF
Geometric	Modern (if available): `=GEOM.DIST(k, p, FALSE)` Legacy: `=GEOMDIST(k, p, FALSE)` Always works (via NegBin, r=1): `=NEGBINOM.DIST(k-1, 1, p, FALSE)`	Modern (if available): `=GEOM.DIST(k, p, TRUE)` Legacy: `=GEOMDIST(k, p, TRUE)` Always works (via NegBin, r=1): `=NEGBINOM.DIST(k-1, 1, p, TRUE)`
Negative Binomial	`=NEGBINOM.DIST(k - r, r, p, FALSE)`	`=NEGBINOM.DIST(k - r, r, p, TRUE)`